$$\mathcal H_c = \mathcal H_S$$

Equation 11 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

A coarse description reports less. It is built from a completely positive, trace-preserving map $\Lambda$ , a quantum channel sending fine states to coarse states, together with a coarse Hilbert space $\mathcal H_c$ and a Hermitian coarse Hamiltonian $H_c$ on it. The simplest case is the partial trace over the bath, $\Lambda(\rho)$ = $\mathrm{Tr}_B$[$\rho$] with $\mathcal H_c$ = $\mathcal H_S$ ; nothing below depends on that particular choice, and a passive relabeling built from no discarding at all, treated next, is a channel too.

Meanings in this article

• $H_c$: wrong, does not touch the total energy of any closed system, and does not by itself explain where a nonzero reading comes from.

Equation guide → · Article →

Equation 58 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Return to the genuine coarse-graining set aside earlier: $\Lambda(\rho)$ = $\mathrm{Tr}_B$[$\rho$] , $\mathcal H_c$ = $\mathcal H_S$ , $H_c$ = $H_S$ . The dual embeds a system observable back into the full space without touching the bath, $\Lambda^\dagger(H_S)$ = $H_S$ $\otimes$ $I_B$ , so

Equation guide → · Article →